Non-uniqueness of weak solutions for the fractal Burgers equation

The notion of Kruzhkov entropy solution was extended by the first author in 2007 to conservation laws with a fractional laplacian diffusion term; this notion led to well-posedness for the Cauchy problem in the $L^\infty$-framework. In the present paper, we further motivate the introduction of entropy solutions, showing that in the case of fractional diffusion of order strictly less than one, uniqueness of a weak solution may fail.Comment: 23 page

On the propagation of a periodic flame front by an arrhenius kinetic

We consider the propagation of a flame front in a solid medium with a periodic structure. The model is governed by a free boundary system for the pair" temperature-front. "The front's normal velocity depends on the temperature via a (degenerate) Arrhenius kinetic. It also depends on the front's mean curvature. We show the existence of travelling wave solutions for the full system and consider their homogenization as the period tends to zero. We analyze the curvature effects on the homogenization and obtain a continuum of limiting waves parametrized by the limiting ratio "curvature coefficient/period." This analysis provides valuable information on the heterogeneous propagation as well.Comment: 42 pages. The statements of Theorems 7 and 8 have been improve

Entropy Solution Theory for Fractional Degenerate Convection-Diffusion Equations

We study a class of degenerate convection diffusion equations with a fractional nonlinear diffusion term. These equations are natural generalizations of anomalous diffusion equations, fractional conservations laws, local convection diffusion equations, and some fractional Porous medium equations. In this paper we define weak entropy solutions for this class of equations and prove well-posedness under weak regularity assumptions on the solutions, e.g. uniqueness is obtained in the class of bounded integrable functions. Then we introduce a monotone conservative numerical scheme and prove convergence toward an Entropy solution in the class of bounded integrable functions of bounded variation. We then extend the well-posedness results to non-local terms based on general L\'evy type operators, and establish some connections to fully non-linear HJB equations. Finally, we present some numerical experiments to give the reader an idea about the qualitative behavior of solutions of these equations

A non-monotone conservation law for dune morphodynamics

26 pInternational audienceWe investigate a non-local non linear conservation law, first introduced by A.C. Fowler to describe morphodynamics of dunes, see \cite{Fow01, Fow02}. A remarkable feature is the violation of the maximum principle, which allows for erosion phenomenon. We prove well-posedness for initial data in $L^2$ and give explicit counterexample for the maximum principle. We also provide numerical simulations corroborating our theoretical results

Continuous dependence estimates for nonlinear fractional convection-diffusion equations

We develop a general framework for finding error estimates for convection-diffusion equations with nonlocal, nonlinear, and possibly degenerate diffusion terms. The equations are nonlocal because they involve fractional diffusion operators that are generators of pure jump Levy processes (e.g. the fractional Laplacian). As an application, we derive continuous dependence estimates on the nonlinearities and on the Levy measure of the diffusion term. Estimates of the rates of convergence for general nonlinear nonlocal vanishing viscosity approximations of scalar conservation laws then follow as a corollary. Our results both cover, and extend to new equations, a large part of the known error estimates in the literature.Comment: In this version we have corrected Example 3.4 explaining the link with the results in [51,59

Fractional semi-linear parabolic equations with unbounded data

International audienceThis paper is devoted to the study of semi-linear parabolic equations whose principal term is fractional, i.e. is integral and eventually singular. A typical example is the fractional Laplace operator. This work sheds light on the fact that, if the initial datum is not bounded, assumptions on the non-linearity are closely related to its behavior at infinity. The sub-linear and super-linear cases are first treated by classical techniques. We next present a third original case: if the associated first order Hamilton-Jacobi equation is such that perturbations propagate at finite speed, then the semi-linear parabolic equation somehow keeps memory of this property. By using such a result, locally bounded initial data that are merely integrable at infinity can be handled. Next, regularity of the solution is proved. Eventually, strong convergence of gradients as the fractional term disappears is proved for strictly convex non-linearity

Occurence and non-appearance of shocks in fractal Burgers equations

International audienceWe consider the fractal Burgers equation (that is to say the Burgers equation to which is added a fractional power of the Laplacian) and we prove that, if the power of the Laplacian involved is lower than 1/2, then the equation does not regularize the initial condition: on the contrary to what happens if the power of the Laplacian is greater than 1/2, discontinuities in the initial data can persist in the solution and shocks can develop even for smooth initial data. We also prove that the creation of shocks can occur only for sufficiently ``large'' initial conditions, by giving a result which states that, for smooth ``small'' initial data, the solution remains at least Lipschitz continuous

Two classes of nonlocal Evolution Equations related by a shared Traveling Wave Problem

We consider reaction-diffusion equations and Korteweg-de Vries-Burgers (KdVB) equations, i.e. scalar conservation laws with diffusive-dispersive regularization. We review the existence of traveling wave solutions for these two classes of evolution equations. For classical equations the traveling wave problem (TWP) for a local KdVB equation can be identified with the TWP for a reaction-diffusion equation. In this article we study this relationship for these two classes of evolution equations with nonlocal diffusion/dispersion. This connection is especially useful, if the TW equation is not studied directly, but the existence of a TWS is proven using one of the evolution equations instead. Finally, we present three models from fluid dynamics and discuss the TWP via its link to associated reaction-diffusion equations

The discontinuous Galerkin method for fractional degenerate convection-diffusion equations

We propose and study discontinuous Galerkin methods for strongly degenerate convection-diffusion equations perturbed by a fractional diffusion (L\'evy) operator. We prove various stability estimates along with convergence results toward properly defined (entropy) solutions of linear and nonlinear equations. Finally, the qualitative behavior of solutions of such equations are illustrated through numerical experiments

Dirichlet forms and semilinear elliptic equations with measure data

We propose a probabilistic definition of solutions of semilinear elliptic equations with (possibly nonlocal) operators associated with regular Dirichlet forms and with measure data. Using the theory of backward stochastic differential equations we prove the existence and uniqueness of solutions in the case where the right-hand side of the equation is monotone and satisfies mild integrability assumption, and the measure is smooth. We also study regularity of solutions under the assumption that the measure is smooth and has finite total variation. Some applications of our general results are given.Comment: Typos corrected. Two examples adde